Forcing Chain

A candidate that forces a contradiction can be ruled out (Nishio).

A forcing chain is a proof by contradiction. You tentatively drop one candidate into a cell, then follow only the moves it forces — the naked and hidden singles that cascade out from that one assumption. If the chain of forced steps drives some cell into an impossible state, such as being left with no candidates at all, the candidate you assumed is proven wrong and can be eliminated. The forcing chain sudoku technique turns a single well-chosen assumption into rigorous logic, because a candidate that leads only to contradiction cannot be the true digit.

Worked example

Forcing chain (Nishio): R1C1 is {1,2} while R1C2 and R4C1 are both forced to 2. Tentatively placing 2 at R1C1 leaves R1C2 with no candidates — a contradiction — so R1C1 must be 1.1222345678945678912378912345631564897564897231897231564312645978645978312978312645
Forcing chain (Nishio): R1C1 is {1,2} while R1C2 and R4C1 are both forced to 2. Tentatively placing 2 at R1C1 leaves R1C2 with no candidates — a contradiction — so R1C1 must be 1.

Here R1C1 holds the pair {1,2}, while its row-1 neighbour R1C2 has already been reduced to a lone 2 (as has R4C1). Tentatively place a 2 in R1C1 and follow what it forces: a 2 in R1C1 removes 2 from R1C2, since the two cells share row 1. But 2 was R1C2's only surviving candidate, so R1C2 is now empty — an impossible state. That contradiction proves R1C1 cannot be 2, so it must take its other candidate, 1.

When to use it

Reach for a forcing chain only after the pattern-based methods have stalled and no single or fish is left to find. Scan for a bivalue cell — one holding exactly two candidates — sitting next to cells already pinned to a single digit, as R1C1 sits in row 1 beside the forced 2 in R1C2. Test one of its candidates, propagate only the forced singles, and watch for a cell emptied of candidates or a digit forced twice into the same unit.

How to use it

  1. Pick a candidate to test. Choose a cell with two or more candidates and tentatively assume one of them is the answer for that cell.
  2. Follow the forced singles. Fill in every move that assumption forces — naked singles and hidden singles — one after another, as far as the chain runs.
  3. Rule it out on a contradiction. If the chain ever empties a cell of all candidates, or leaves a unit with no home for a digit, the assumption was wrong — remove that candidate from the starting cell.
Practice on a real puzzle

Frequently asked questions

What is a Forcing Chain in Sudoku?

A forcing chain is a proof by contradiction: you assume one candidate is true, propagate every naked and hidden single it forces, and if the grid collapses into an impossibility you eliminate that candidate. In the example, assuming R1C1 is 2 empties R1C2, so 2 is ruled out and R1C1 must be 1. Nothing is guessed — the discarded digit is shown to be logically impossible, and the other candidate is confirmed as the answer.

How do I spot a Forcing Chain on the grid?

Look for a bivalue cell — one with exactly two candidates — that borders cells already forced to a single digit in the same row, column, or box. That closeness keeps the propagation short and easy to verify by eye. Here R1C1's pair {1,2} shares row 1 with R1C2, which is already down to a lone 2, so testing the 2 in R1C1 collides almost immediately.

Is a Forcing Chain just guessing?

No. A guess accepts a placement and hopes it holds; a forcing chain assumes a candidate specifically in order to disprove it. Because assuming 2 in R1C1 forces R1C2 to be left empty, that 2 is not merely untried but genuinely impossible, so the elimination is a sound logical deduction. The surviving candidate, 1, is proven rather than gambled on.